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H: Banach space, Normed vector space
Help me please with this question.
Let's $Y$ be Banach space, $Z$ - Normed vector space and $(T_{n})_{\mathbb{N}}$ - the sequence in $B(Y,Z)$ so that all sequence $(y_{n})_{\mathbb{N}}$ in Y holds:
if $\left \| y_n \right \|_{n \to \infty }\rightarrow 0$ then $\left \| T_ny_n \rig... |
H: Continuity of Multivariable function
While doing revision I stumbled upon this problem:
Is this function continuous at the origin?
$$
f_3(x,y) =
\begin{cases}
\frac{x^3-y^3}{x^2+y^2}, & \text{if }(x,y)\not= (0,0) \\
0, & \text{if } (x,y)=(0,0) \end{cases}
$$
The answer is yes, but how do I prove it?
Sincere thank... |
H: a question on distributions.
Suppose $L\in \mathcal{S}$ is a tempered distribution, and for each $\phi \in \mathcal{S} :\phi \geq 0 \implies L(\phi) \geq 0$.
Prove that there exists a borel measure $\mu$ with polynomial growth such that $L=L_{\mu}$, where $L_{\mu}(\phi) = \int \phi \:d\mu$.
I think I need to use he... |
H: Characteristic equation for 2-nd order ODE
Given a differential equation $\dot x = Ax$, $x \in \mathbb{R}^n$ we define its characteristic equation as $\chi(\lambda) = \det (\lambda I - A)$. Consider now the second order ODE
$$
\ddot x + A x + B \dot x = 0, \;\;\; x \in \mathbb{R}^n
$$
With substitution $u = x$, $... |
H: How can an ordinary function be a distribution?
I think distributions are linear and continuous functionals on the set of testfunctions. In a textbook I saw this question:
Let $f$ be a $2\pi$-periodic function with $f(t) = \frac{\pi}{4}|t|$
for $t \in [-\pi, \pi]$, show that it is a distribution!
How could this... |
H: Slope of curve in $\mathbb{R}^3$
While doing revision, I came across this problem:
The surface given by $z=x^2-y^2$ is cut by the plane given by $y=3x$, producing a curve in the plane. Find the slope of this curve at the point $(1,3,-8)$.
I tried substituting $y=3x$ into $z=x^2-y^2$, yielding $z=-8x^2$. Then, $\fra... |
H: Does this sequence converge to $\pi$?
I have a problem with the following sequence
$$ \lim_{n \to \infty} g_n \stackrel{?}{=} \pi $$
where
$$g_n = \sum_{k=1}^{n-1} \frac{\sqrt{\frac{2n}{k}-1}}{n-k}
+ \sum_{k=n+1}^{2n-1}\frac{\sqrt{\frac{2n}{k}-1}}{n-k}.$$
Does it converge to $\pi$? I tested experimentally that it
... |
H: Solving $\sin (5\phi)-\sin \phi=\sin (2\phi)$
The question is :
Find the general solution of this equation
$$
\sin (5\phi)-\sin \phi=\sin (2\phi)
$$
I tried to expand $\sin (5\phi)$ and $\sin (2\phi)$,
so the equation only contains $\cos\phi$ and $\sin \phi$.
But I can't make it into a form like $\sin(\phi+a)=n$
to... |
H: Accumulation points / Cluster points / Closed sets
In a topological space $X$, call $x\in X$ an accumulation point if $\forall$ open set $U\ni x$, $U \cap A \neq \emptyset$, and $y\in X$ a cluster point if $\forall$ open set $U\ni y$, $U\cap A\setminus \{y\} \neq \emptyset$. (These are the terminologies used by my ... |
H: Prove that $K$ is a field
Let be $K$ the set of real numbers that can be written as $a+b\sqrt2$, with $a$ and $b$ rational numbers. Prove that $K$ is a field.
I have already proved that $0$ and $1$ $\in K$, and that sum and product of two elements $\in K$. I have also already proved that the opposite $\in K$. I don... |
H: Logic question proving something about compactness
Let $\Sigma$ be a set of formulas. There's a finite set $\Lambda \subseteq \Sigma$.
I'm asked to prove or disprove that $\Sigma$ has a model if and only if $\Lambda$ has a model.
It seems to me it's just true using compactness, but that sounds too easy.
AI: It is ... |
H: Discontinuous optimizer but continuous optimal
Consider a locally-bounded, continuous, positive-semidefinite function $f: X \times Y \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact, $Y \subseteq \mathbb{R}^m$.
For each $y \in Y$, define:
$$ f^*(y) := \min_{x \in X} f(x,y) $$
$$ x^*(y) :=... |
H: Intuition for the following change of index of summation
I am working through Concrete Mathematics. I came across the following change in index of summation while going through the number theory chapter.
$$\sum_{m|n}^{ } \sum_{k|m}{ } a_{k,m} = \sum_{k|n}^{ } \sum_{l|(n/k)} a_{k,kl}$$
If I list out the complete su... |
H: Closed form expression for the k-th term of a sequence.
Let
$$
x_{n+1} = \left\{ \begin{array}{c} x_{n}^2 & \mbox{if $b_{n+1}=1$} \\ \alpha x_n & \mbox{if $b_{n+1}=0$}\end{array} \right. ,
$$
for $n\ge 0$ and $\alpha > 1$. Can we write $x_k$ in terms of $x_0$ and $P$ where
$$
P = \sum_{i=1}^{k} b_i.
$$
For $\alph... |
H: how to rotate a Gaussian?
Lets suppose that we have a 2D Gaussian with zero mean and one covariance and the equation looks as follows
$$f(x,y) = e^{-(x^2+y^2)}$$
If we want to rotate in by an angle $\theta$, does it mean that we rotate the values $x$ and $y$ and then see how the Gaussian is rotated or do we actuall... |
H: If a function has a finite limit at infinity, does that imply its derivative goes to zero?
I've been thinking about this problem: Let $f: (a, +\infty) \to \mathbb{R}$ be a differentiable function such that $\lim\limits_{x \to +\infty} f(x) = L < \infty$. Then must it be the case that $\lim\limits_{x\to +\infty}f'(x... |
H: Vector Identity and the Scalar Product
Ok, I have a question that probably has a very simple answer but for some reason I can't see it. Let $a$ and $r$ be two vectors of nonzero length with a common origin and let $\theta$ be the nonzero angle between them. Then, by definition of the cosine funtion,
$$
\cos \theta ... |
H: About Kirby Diagrams
I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. There is something I don't understand on page 116 (Google Books link to page 116; alternatively, here are images of page 115 and page 116)
Now, we consider compact 4-manifolds, and they have handle decomposition.
0-handle... |
H: If $ \frac{1}{c} = \frac{1}{2a} + \frac{1}{2b} $, then either $ a \leqslant c \leqslant b$ or $ b \leqslant c \leqslant a $
For $a, b = 1, 2, 3, \cdots$, let $ \frac{1}{c} = \frac{1}{2a} + \frac{1}{2b} $. Then prove that either $ a \leqslant c \leqslant b$ or $ b \leqslant c \leqslant a $ holds.
AI: Either $\frac{1... |
H: Product of varieties
If we have two rational varieties (i.e varieties which are birational to some projective space) is their product also a rational variety? would this rely on the fact that the Zariski topology is finer than the product topology and that $\mathbb{P}^{r} \times \mathbb{P}^{s}$ is birational to $\m... |
H: Infinite intersection of kernels
Let $X$ be an infinite dimensional Banach space. And let $X^{\mathrm{*}}$ be the space of linear continuous functionals( $f : X \rightarrow \mathbb{R}$ linear and continuous). Assume $X^{\mathrm{*}}$ is separable, and take $(x^{\mathrm{*}}_n)$for $n \in \mathbb{N}$ be a dense subset... |
H: Computing roots of high degree polynomial numerically.
Here is my problem ; for my research, I believe that the complex numbers I am looking at are precisely the (very large) set of roots of some high degree polynomial, of degree $\sim 2^n$ where $1 \le n \le 10 \sim 15$. Mathematica has been running for the whole ... |
H: Evaluating $\int{\frac{x}{1+x^4}\ dx}$
I have this integral:
$$\int{\frac{x}{1+x^4}\ dx}$$
The solution should be:
$$\frac{1}{2} \arctan{x^2}+C$$
But I have only seen how to integrate when in denominator I have an expression with real roots. Here, with $1+x^4$ I dont have any real roots and I don't know how I can i... |
H: Why do irrationality proofs of $\sqrt x$ not apply when $x$ is a perfect square?
When trying to prove that a particular root (say $\sqrt{2}$ or $\sqrt{10}$) cannot be rational, I always see a particular indirect proof that goes something like this:
Suppose $\sqrt{x}$ were rational; then, there would be two integers... |
H: PEMDAS:How to solve this exercise?
I have the following problem:
$$100*\left\{24+100*[1001-4*(25*6+25*4)]\right\}$$
I'm very frustrated that I can't solve exercises like this. I have read about PEMDAS and followed the steps but somehow I am making a mistake because the result I get is not correct.(looked at answers... |
H: A question on modules over Noetherian ring
If $G$ is a module over the non-trivial commutative Noetherian ring $R$ then is it possible that for all maximal ideal $M$ of $R$ we have $MG=G$ ?
I guess the answer is no.
AI: Actually the answer is yes:
$R=\mathbb Z, G=\mathbb Q$ . |
H: Velocity Question & Acceleration
Below I have a question that I tried to solve on a exam. I am curious as to the actual way to approach the question. What I did was set the equation equal to $0$ and get $t = -3$ then I plugged in $3$ into the derivative of the equation and I got $128$ ft/sec. But apparently I appro... |
H: Techniques of Integration w/ absolute value
I cannot solve this integral.
$$
\int_{-3}^3 \frac{x}{1+|x|} ~dx
$$
I have tried rewriting it as:
$$
\int_{-3}^3 1 - \frac{1}{x+1}~dx,
$$
From which I obtain:
$$
x - \ln|x+1|\;\;\bigg\vert_{-3}^{\;3}
$$
My book (Stewart's Calculus 7e) has the answer as 0, and I can intuit... |
H: Is this proof that $\sqrt 2$ is irrational correct?
Suppose $\sqrt 2$ were rational. Then we would have integers $a$ and $b$ with $\sqrt 2 = \frac ab$ and $a$ and $b$ relatively prime.
Since $\gcd(a,b)=1$, we have $\gcd(a^2, b^2)=1$, and the fraction $\frac{a^2}{b^2}$ is also in lowest terms.
Squaring both sides,
... |
H: Compute the limit of $\frac1{\sqrt{n}}\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)^{1/n^2}$
Compute the following limit:
$$\lim_{n\to\infty}\frac{{\left(1^1 \cdot 2^2 \cdot3^3\cdots n^n\right)}^\frac{1}{n^2}}{\sqrt{n}} $$
I'm interested in almost any approaching way for this limit. Thanks.
AI: Let's begin
$$
\lim... |
H: Fundamental Group of the complement of $\mathbb{S}^1$ union with $z$ axis in $\mathbb{R}^3$
Can you help me to compute the fundamental group of $\mathbb{R}^3\setminus X$, where $$X = \mathbb{S}^1\cup \{(0,0,z)\in\mathbb{R}^3\mid z\in\mathbb{R}\}\ ?$$
I'm sorry if I repeated the question and I'm sorry if the questi... |
H: Method of Frobenius
Can someone please explain the green text to me?
Maybe I am not reading it right, but that sentence makes no sense to me. "that (2) is an equation For which $xp(x)$ and $x^2q(x)$ " what does this part mean? The next part that follows sys something about being constants. Are they trying to say w... |
H: proof of inequality-using a convex function
Let $a_{1},a_{2},\ldots,a_{n},b_{1},b_{2},\ldots,b_{n}$ be positive numbers. We need to prove that:
$$(a_{1}+b_{1})^{\frac{1}{n}}(a_{2}+b_{2})^{\frac{1}{n}}\cdots(a_{n}+b_{n})^{\frac{1}{n}}\geq a_{1}^{\frac{1}{n}}a_{2}^{\frac{1}{n}} a_{n}^{\frac{1}{n}}+b_{1}^{\frac{1}{n... |
H: Help Verifying Trigonometric Identity
I could really use help, hint or otherwise, in proving a trigonometric identity:
We are only allowed to work on one side of the equation.
$$\dfrac{2\sin^2(x)-5\sin(x)+2}{\sin(x)-2} = 2\sin(x)-1$$
AI: HINT: Factorize the numerator and cancel terms arguing why the terms you are ... |
H: Bounds for $\zeta$ function on the $1$-line
I was going over my notes from a class on analytical number theory and we use a bound for the $\zeta$ function on the $1$ line as $\vert \zeta(1+it) \vert \leq \log(\vert t \vert) + \mathcal{O}(1)$ for $t$ bounded away from $0$, say $\vert t \vert \geq 1$. I don't seem t... |
H: Does the curvature determine the metric?
Here I asked the question whether the curvature deterined the metric. Since I am unfortunately completely new to Riemannian geometry, I wanted to ask, if somebody could give and explain a concrete example to me, as far as the following is concerned:
At the MO page (as cited ... |
H: Does the series converge or diverge?
Let
$$
a_n=\frac{\sqrt{3n+1}}{n^2}
$$
I cannot find a suitable $b_n$ to use for the comparison test, and when I try to use the ratio test it really becomes a mess and I cannot find the limit as a result. I am going to guess there is something simple I am missing.
AI: I am assumi... |
H: Proof an inequality
I'm trying to prove that
$$ \frac{3-2\sqrt{1-15 m^2}}{1+12 m^2}\geq 1+3 m^2$$
I have obtained in a CAS software the Taylor expansion in $m=0$
One posibility to prove the inequality is showing coeficients in Taylor expansion are non-negative, by I don't find how.
Really I want only to obtain ... |
H: Spanning tree of a strongly connected directed graph.
Given a strongly connected directed graph $G=(V,E)$, and a node $r \in V$. Let $T_r$ be the set of spanning trees of $G$ with $r$ as root and all edges pointing to $r$. Is is it possible that there is an edge $e$ such that for all $t \in T_r$, we have $e \in E... |
H: Every Hilbert space operator is a combination of projections
I am reading a paper on Hilbert space operators, in which the authors used a surprising result
Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections.
The author referred to a 1967 paper by Fillmore, Sums of opera... |
H: Integral of $\int{\frac{dx}{(\arcsin{x})\sqrt{1-x^2}}}$
I am having a problem solving an integral. I am stuck in an infinite loop. Integral is:
$$\int{\frac{dx}{\sqrt{1-x^2}\arcsin{x}}}$$
I have separated it in dv and u on this way:
$$u = \frac{dx}{\sqrt{1-x^2}}$$
$$dv = \frac{1}{\arcsin{x}}$$
And the using:
$$u v ... |
H: Linear algebra: Inverse of a matrix?
How does $$\mathbf P^{-1}\mathbf A \cdot \mathbf I \cdot \mathbf P = \mathbf A\mathbf P^{-1}\cdot \mathbf P = \mathbf A\cdot \mathbf I\ ?$$
I thought $\mathbf P^{-1}$ couldn't be "moved around" within an equation.
It's from a theorem that similar matrices have the same eigenva... |
H: How to compute $\int\frac{x^2}{x^2+4}\,dx$ and $\int\frac{1}{\sqrt{4-x^2}}\,dx$?
My calculus teacher gave us a list of integrals to solve through the substitution method. I've been trying everything for hours, and I just can't find out how to compute these:
$$∫\frac{x^2}{x^2+4}\,dx\qquad\text{and}\qquad \int\frac{1... |
H: Regarding direction of gradient of a function
I was reading this book related to optimizing a function f(x,y) with constraints g(x,y) = 0. The book says that let us suppose g(x,y) define a surface, then gradient of the g(x,y) will be orthogonal to the surface. But I am a bit confused how is it possible. I mean grad... |
H: Trying to rederive an exponential approximation
So I was reading a paper where the following approximation was made. Note that $p$ is small, $L$ is large, and $pL$ is $O(1)$:
$$\left[1-e^{-4p(1+p)L}\right]^{L/2}=\textrm{exp}\left[\frac{2(pL)^2}{e^{4pL}-1}\right](1-e^{-4pL})^{L/2}+O(p^3L^2)$$
I just don't see where ... |
H: Maximum and Minimum Perimeter of a Triangle
I cant figure out the following question:
A triangle has sides with lengths of 9,14 and h. if h is an integer what is the difference between the maximum and minimum possible perimeter of the rectangle ? (Ans=17.5)
Any suggestions on how this should be solved ??
AI: No... |
H: Linear algebra: diagonalization
Let $L: P_2 \longrightarrow P_2$ be the linear operator defined by $L(p(t)) = p'(t)$ for $p(t) \in P_2$, the space of real polynomials of degree at most $2$. Is $L$ diagonalizable? If it is, find a basis $S$ for $P_2$ with respect to which $L$ is represented by a diagonal matrix.
Ans... |
H: Evaluating a double integral with change of coordinates. $\int\!\!\int_{R} (x+y)\,dA$
It looks simple, but I'm having a bit of difficulty with trying to obtain the proper change of coordinates.
Evaluate the following integral
$$\displaystyle\int\!\!\!\int_{R} (x+y)\,dA$$
with region R in the first quadrant boun... |
H: Is $z=x^2+y^2$ a bijection?
I am learning basic set theory and was doing some exercises where we are to determine whether a given relation is a function, an injection, a surjection and if it is a bijection.
The question in this case was: Determine whether the relation from $\mathbb{R}^2$ to $\mathbb{R}$ defined by... |
H: Line integral with only one unknown
Let $\Gamma$ be a circumference with center $3i$ and radius $5$
oriented counter-clockwise. Calculate: $$\displaystyle\oint_{\Gamma} \frac{z}{(z^2 -2z)(z^2 - 4z + 13)} dz$$
At a first glance I thought it was a line integral but then I realized the function $f(z)$ has only one... |
H: Generating the Sorgenfrey topology by mappings into $\{0,1\}$, and on continuous images of the Sorgenfrey line
Show that the topology of the Sorgenfrey line can be generated be a family of mappings into a two-point discrete space.
Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$ but cannot be mappe... |
H: Compute: $\lim_{n\to\infty} ({x_{2}}^n+{x_{3}}^n)$
Let be $ x_{1}, x_{2}, x_{3}$ the roots of $x^3-x^2-1=0$. If $x_{1}$ is the real root of the equation, then calculate:
$$\lim_{n\to\infty} ({x_{2}}^n+{x_{3}}^n)$$
I wonder if this limit can be computed without being necessary to know the exact values of $x_{2}$ an... |
H: Finding all integer solutions for $x^2 - 2y^2 =2 $
I'd love your help with finding all the integer solutions to the following equation:
$x^2 - 2y^2 =2 $. I want to use Pell's theorem so I changed the equation to $-\frac{1}{2}x^2+ y^2 =-1$, Can I use Pell's Theorem now? I got a private solution for $-\frac{1}{2}x^2+... |
H: Eigenvectors of a matrix and its diagonalization
I'm trying to understand the relation (if any) between the eigenvectors of similar matrices and in particular of a matrix and its diagonalization.
Given $A,D\in M^F_{n\times n}$ and invertible $P$ such that $P^{-1}AP=D$ then $AP=PD$ and the eigenvectors of A are the ... |
H: What is a formal definition of series?
Is there a formal definition for series?
For example, cardinal sum has a formal definition such that $\sum a_i$ = $\bigcup a_i$.
Is there any clear definition for series of real or complex number?
The definition on my book is the sum of $a_0 + ... + a_n$. This seems very intui... |
H: A problem on joint distribution
Suppose that the joint distribution of $X$ and $Y$ is uniform over the region in the $xy$-plane bounded by $x=-1,x=1,y=x+1, \text{ and }y=x-1$.
What is $\mathbb{P}(XY>0)$?
What is the conditional p.d.f. of $Y$ given that $X=x$?
AI: The region in the $XY$-plane is as shown below.
HI... |
H: Preimage of a point by a non-constant harmonic function on $\mathbb{R}$ is unbounded
Let $u$ be a non-constant harmonic function on $\mathbb{R}$. Show that $u^{-1}(c)$ is unbounded.
I am not getting what theorem or result to apply. Could anyone help me?
AI: Let $u(x)=x$, for all $x \in \mathbb{R}$. Then $u''(x)=0... |
H: Construction a sequence of real numbers
Can we construct a sequence $\{a_{i}\}$, where $0<a_{i}\leq 1$ for all $i$, such that $\sum_{i=1}^{n}a_{i}=\frac{B_{n}}{2^{n}}$ such that $B_{n}\to b$ as $n\to \infty$ for some $1\leq b <\infty$?
Edit: What about $\sum_{i=1}^{n}a_{i}=\frac{n}{n+1}$, or $\sum_{i=1}^{n}a_{i}=... |
H: Counting number of ways of splitting a card deck
Suppose that we have a $52$-card deck. We are interested to find how many different combinations there could be if we divide this $52$-card deck in two parts, so that in each part there are $2$ aces.
What I am thinking is that we have $4$ aces, we can choose two of t... |
H: System of Parameters.
Let $R=k[X,Y,U,V]/(XV-YU)$, where $k$ is field of characteristic $0$. Consider $S=R_m$, where $m$ is the maximal ideal $(X,Y,U,V)/(XV-YU)$. How can we find a system of parameters for $S$ and what are they? We know that there are 3 elements in any system of parameters, as $S$ is a 3-dimensional... |
H: Polynomial of same degree
Let $p(z)$ and $q(z)$ are two polynomial of same degree and zeroes of $p(z)$ and $q(z)$ are inside open unit disc, $|p(z)|=|q(z)|$ on unit circle then show that $p(z)=\lambda q(z)$ where $|\lambda|=1$.
Please just give a hint not the whole solution. Thank you.
AI: Let $d$ the common degr... |
H: Unique isomorphism between fields generated by a domain.
Suppose $F$ and $K$ are fields both generated by a common subring $D$, which is a domain. My question is, why is there a unique isomorphism between $F$ and $K$ which is the identity on $D$?
Wouldn't the field generated by $D$ be unique? So $F=K$, and then any... |
H: Modular arithmetic for negative numbers
If I have the congruence
$$m^2 \equiv -1 \pmod {2k+1}$$
how do I solve for the solutions to this congruence (given that I know $k$)?
AI: As others have pointed out, when dealing with congruences the concept of a negative number is meaningless (as is the concept of a positive ... |
H: $|G|=12$ and no elements of order $2$ in $Z(G)$
I am thinking on the following problem:
If $|G|=12$ and there is no element of order $2$ in its center then $3$-Sylow subgroup of $G$ cannot be normal in $G$.
I was told that to assume the $3$-Sylow subgroup of $G$, say $P$, is normal in the group and go to reach a ... |
H: How to show the dimensionality of a tangent space defined via equivalence classes of curves?
Lets have smooth $n-$ manifold $M\subset R_N$ and define its tangent space at $p\in M$ to be set of equivalence classes of smooth curves with $\gamma: I\to M$, $\gamma(0)=p$ with relation $\gamma_1\sim\gamma_2$ if $\frac{d\... |
H: Normal subgroups of p-groups
Let $G$ be a group of order $p^\alpha$, where $p$ is prime. If $H\lhd G$, then can we find a normal subgroup of $G/H$ that has order $p$?
AI: Theorem: a group $\,G\,$ of order $\,p^n\,,\,p\,$ a prime, $\,n\in\mathbb N\,$ , always has normal subgroup of order $\,p^m\,\,,\,\,\forall\, m\l... |
H: Solving $\sin 7\phi+\cos 3\phi=0$
The question is find the general solution of this equation:$$\sin(7\phi)+\cos(3\phi)=0$$
I tried to use the "Sum-to-Product" formula, but found it only suitable for $\sin(a)\pm \sin(b)$ or $\cos(a)\pm \cos(b)$. So I tried to expand $\sin 7\phi$ and $\cos 3\phi$, but the equation... |
H: Finding a proper sequence
The following function was given to me $$f(x)=\lfloor x\rfloor+\lfloor-x\rfloor$$
wherein $\lfloor x\rfloor$ is the floor function of $x$. I was asked to select a proper sequence for showing that this function has no limit at $\infty$.
Honestly, my knowledge about analysis is weak. Thank y... |
H: How to denote a "boolean" (on/off, 1/0) variable in mathematics?
Is there any conventional notation for variables that can only take the value 0 or 1? (I'm looking for something of the nature of an overbar, a caret, etc.)
AI: I don't know of such notation. You can always define that $\dot x$ means that $x$ is a Boo... |
H: The congruence $f(x)=x^3+3x+9 \equiv 0 (\bmod 5^n)$ has only one solutions for every $n \geq 2$
I need to prove that the congruence $f(x)=x^3+3x+9 \equiv 0 (\bmod 5^n)$ has only one solutions for every $n \geq 2$.
I checked with Hensel theorem that for $n=2$ there is one solution indeed.
I want to use induction a... |
H: how to change polar coordinate into cartesian coordinate using transformation matrix
I would like to change $(3,4,12)$ in $xyz$ coordinate to spherical coordinate using the following relation
It is from the this link. I do not understand the significance of this matrix (if not for coordinate transformation) or how... |
H: Finding the set of solutions
Kindly asking to find the set of possible solutions if they exist of the equation $$\lfloor x+\lfloor x+\lfloor x\rfloor\rfloor\rfloor+3\lfloor x\rfloor=18$$ Of course I have 4 intervals as choices:
$[\frac{5}{2}, \frac{7}{2})$
$[3, 5)\smallsetminus \{4\}$
$[3, 4)$
$\emptyset$
$\lfl... |
H: Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
AI: Yes. For every $r\in\mathbb R$ choose a sequen... |
H: Definition and meaning of "Proof Schema", "Class Sign"
I'm a newbie in advanced mathematics, and I'm trying to understand Godel's theorem. I came across these two words which I couldn't understand clearly.
"Proof Schema" and "Class-Sign"
Can anybody provide me definition of these, and describe what these term mean ... |
H: How can I calculate this limit: $\lim\limits_{x\rightarrow 2}\frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}}$?
I was given this limit to solve, without using L'Hospital rule. It's killing me !! Can I have the solution please ?
$$\lim_{x\rightarrow 2}\frac{2-\sqrt{2+x}}{2^{\frac{1}{3}}-(4-x)^\frac{1}{3}}$$
A... |
H: How is $\Vert x \Vert_3$ constrained by $\Vert x \Vert_2$ and $\Vert x \Vert_4$ for $x\in \ell^2$?
It seems we can find some $x\in \ell^2$ with $\Vert x \Vert_2=1$ that has $\Vert x \Vert_4=a$ for any $0<a\le 1$.
But can we find an $x$ with $\Vert x \Vert_2=1,\Vert x \Vert_3=b,\Vert x \Vert_4=a$ for every choice of... |
H: Show $|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$ for complex $a$, $b$, $c$
How to prove for any complex numbers $a$, $b$, $c$, the inequality $$|a|+|b|+|c|+|a+b+c| \geq |a+b|+|b+c|+|c+a|$$ is correct?
AI: Both sides are non-negative, so it suffices to show that the square of the left-hand-side is at least the sq... |
H: Does $\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}$ converge?
Does the following series converge or diverge?
$$
\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}
$$
The methods I have at my disposal are geometric and harmonic series, comparison test, limit comparison test, and the ratio test.
AI: It is... |
H: How to integrate $\int{\frac{dx}{\sqrt{16-9x^2}}}$
Me again, probably someone is going to blame me. I have this:
$$\int{\frac{dx}{\sqrt{16-9x^2}}}$$
I have asked an old teacher of mine an he said me I should let $x=\frac{4}{3}\sin{\big(\frac{3}{4}x\big)}$ but I don't how he realise that. I have tried $u-substitutio... |
H: cardinality of the set of countable partitions of $\mathbb{R}$
What is the cardinality of the set
$$A=\{ P| P\ \text{is a countable partition of the reals} \}$$ ?
I am searching on this for a while. I think the cardinality is $2^w$ where $w$ is the cardinality of $\mathbb{N}$. Clearly $|A|\geq |\mathbb{R}|=2^w$, ... |
H: Multiplying by invertible matrices maintains similarity
I know that given any two similar matrices $A,B\in M^F_{n\times n}$ and any two invertible matrices $P,Q$ of the same order then $P^{-1}AP$ is similar to $Q^{-1}BQ$.
However I don't completely understand why. Could someone explain it?
AI: We use the fact th... |
H: For prime $p$, $p\equiv 5 (\bmod 8)$, and $a$ such that $\left(\frac{a}{p}\right)=1$: $a^{\frac{p-1}{4}}$ is either $1$ or $-1$
I'd really love your help with the following problem: For prime $p$, $p= 5 (\bmod 8)$, and $a$ such that $(\frac{a}{p})=1$ (Legendre symbol): I need to show that $a^{(p-1)/4}$is either $1$... |
H: On the Dirichlet beta function sum $\sum_{k=2}^\infty\Big[1-\beta(k) \Big]$
Given the Dirichlet beta function,
$$\beta(k) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^k}$$
(The cases k = 2 is Catalan's constant.) It seems,
$$\sum_{k=2}^\infty\Big[1-\beta(k) \Big] = \frac{1}{4}\big(\pi+\log(4)-4\big)=0.131971\dots$$
or, ... |
H: Is $\log_{5}{-3} = \frac{\log(3)+\pi i}{\log(5)}$?
Why does my calculator return false when I input $\log_{5}{-3} = \frac{\log(3)+\pi i}{\log(5)}$ but W|A returns true?
I'm thinking my calculator is wrong because I know that $\displaystyle \log_{5}{(-3)} = \frac{\log(-3)}{\log(5)}$ and $\displaystyle \log(-x) = \lo... |
H: What does it mean to say an integral exists 'in the distributional sense'?
What exactly does it mean to say that an integral exists 'just in the distributional sense'? For example, the Fourier transform of $x^2 e^{-\lambda x}$ or of $H(R-|x|)$ where $R > 0$ and $H$ is the Heaviside-Function, are often said to exist... |
H: How to evaluate $\displaystyle{\int{\frac{1}{\sqrt{e^{2x}-4}}}\,dx}$
Please, could someone correct my exercise? I got it by myself, but the result that I found is differend by the WolframAlpha result... Could you tell me, please, if I'm wrong an where?
Thanks in advance!
AI: There is an error in the beginning. Se... |
H: a question on one one function on an open subset of $\mathbb{R}^n$
$1$)Let $A\subseteq\mathbb{R}^n$ be an open set and $f:A\rightarrow \mathbb{R}^n$ a continuously differentiable $1-1$ function such that $det f'(x)\neq 0$
for all $x$. Show that $f(A)$ is an open set and $f^{-1}: f(A )\rightarrow A$ is differ
enti... |
H: Is it assumable that $2^{1/12}$ is irrational because $2^{1/2}$ is?
I need to prove that $2^{1/12}$ is irrational but I need to connect this to $2^{1/2}$ being irrational.
I know how to prove that $2^{1/2}$ is irrational, but can I assume that $2^{1/12}$ is irrational because $2^{1/2}$ is and why?
Cheers
AI: If $2^... |
H: Formula for the sequence repeating twice each power of $2$
I am working on some project that needs to calculate what $a_n$ element of the set of numbers $$1, 1, 2, 2, 4, 4, 8, 8, 16, 16 \ldots$$ will be.
$n$ can be quite big number so for performance issues I have to calculate formula for this set of numbers (Note:... |
H: Small categories
Why $R$-Mod is a small category? There is a way to recognize small categories?
For example Grp (i.e. category of all groups) is large because every set can be equiped with a group structure.
AI: For any ring $R$, the category $R$-Mod isn't a small category: for any set $S$ one can form the $R$-mo... |
H: Integral of $\frac{1}{\sin z}$ along a path
Suppose $\gamma$ is a simple, closed path, with $0$ in its interior and $\{\pi n:n\in\mathbb{Z}\setminus\{0\}\}\subset\mathbb{C}\setminus|\gamma|$. Find
$$
\int_{\gamma} \frac{1}{\sin z} dz
$$
Perhaps it's a simple question of Cauchy's formula or Cauchy's theorem.
T... |
H: Some properties of Yosida-Moreau transform
Let $f(x)$ be a continuous function on $\mathbb{R}^n$, $f(x) \geqslant 0$ for any $x$. Define
$$
f_{\alpha}(x) = \inf\limits_{y}\left( f(y) +\frac{|x-y|^2}{2\alpha} \right)
$$
where $\alpha > 0$. How to show that $f_{\alpha}(x)$ is locally Lipschitz continuous and $f_{\... |
H: Integral - using Euler Substitution
I've been trying to solve one simple Integral with Euler substitution several times, but can't find where I'm going wrong. The integral is (+ the answer given here, too):
$$\int\frac{1}{x\sqrt{x^2+x+1}} dx=\log(x)-\log(2\sqrt{x^2+x+1}+x+2)+\text{ constant}$$
The problem is, I can... |
H: Linear algebra: power of diagonal matrix?
Let A = $\begin{pmatrix} 3 & -5 \\ 1 & -3 \end{pmatrix}$. Compute $A^{9}$. (Hint: Find a matrix P such that $P^{-1}AP$ is a diagonal matrix D and show that $A^{9}$= $PD^{9}P^{-1}$
Answer: $\begin{pmatrix} 768 & -1280 \\ 256 & -768 \end{pmatrix}$
I keep getting $\begin{pmatr... |
H: Which theories and concepts exist where one calculates with sets?
Recently I thought about concepts for calculating with sets instead of numbers. There you might have axioms like:
For every $a\in\mathbb{R}$ (or $a\in\mathbb{C}$) we identify the term $a$ with $\{a\}$.
For any operator $\circ$ we define $A\circ B :=... |
H: Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent
I'm new to topology, and can't figure out why the metric and product topologies over $\mathbb{R}^n$ are equivalent. Could someone please show me how to prove this?
AI: The product topology is induced by this norm.
$$\|x\|_{\rm prod} = \m... |
H: Generalisation of a polynomial factorisation
I know of a theorem in algebra, that every polynomial $p\left(X\right)=a_{0}+a_{1}X+\ldots+a_{n}X^{n}$
that is nonconstant and has real coefficients, admits a factorisation
of the form
$$
p\left(X\right)=c\left(X-\lambda_{1}\right)\ldots\left(X-\lambda_{m}\right)\left(X... |
H: Is Frenet-Serret frame valid for non-natural parametrized curves when one normalizes the tangent vector?
When one have a curve $\beta(s)$ which is parametrized by arc length (has natural parametrization) one is able to obtain the tangent, normal and binormal vectors by using Frenet-Serret frame equations:
$T = \bet... |
H: What kind of analytical function could resemble that plot?
I am in the middle of making a model, and I am looking for an analytical expression which could resemble this evolution for one of the parameters:
In short: a sudden increase from 0 to a global maximum, followed by slower decrease. Once I ha... |
H: Riemann zeta sums and harmonic numbers
Given the nth harmonic number of order s,
$$H_n(s) =\sum_{m=1}^n \frac{1}{m^s}$$
It can be empirically observed that, for $s > 2$, then,
$$\sum_{n=1}^\infty\Big[\zeta(s)-H_n(s)\Big] = \zeta(s-1)-\zeta(s)$$
Can anyone prove this is true?
AI: $$\sum_{n=1}^{\infty} (\zeta(s) - H_... |
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